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On the one hand, John M is correct that I was sloppy in stating that
-P dV is the work.
The work is positive but since I'm moving the
piston rapidly it cannot be easily calculated as an integral over the
actual process
but instead should be calculated from W = E_final -
E_initial which we can do because we have well-defined equilibrium
initial and final states.
... I think we *can* have irreversible infinitesimal
processes. It's just that dQ doesn't equal T dS *and* dW doesn't
equal -P dV for them.
... My claim was that
the final temperatures would be the same if the process proposed
would be done quickly or slowly.